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Homework Help: Relativistic Group Velocity Calculation

  1. Oct 29, 2007 #1
    1. The problem statement, all variables and given/known data

    we are given that an electron and a proton have the same KE.We are to compare their phase and group velocity...


    2. Relevant equations



    3. The attempt at a solution

    K=(γ₁-1)m₁c²=(γ₂-1)m₂c²
    Now, I found it very problematic to extract the ratio of v₁/v₂ in terms of m₁ and m₂
    So,I expanded the γ s binomially where the major contribution comes from the first few terms...It follows that group velocity if proton is much less than that of the electron...

    Please tell me if I am correct and sggest any other possible ways...

    Regards,
    neelakash
     
  2. jcsd
  3. Oct 29, 2007 #2

    clem

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    Don't use v.
    Use E=T+M and p=\sqrt{T^2+2M}.
     
  4. Oct 29, 2007 #3
    but how would you compae between the group velocities???

    I hope your formula is pc=√[K(K+2mc²)] where K is the KE

    But, p=γmv...so that you are to know γ if you want to know v
    γ s are different for e and p...
     
  5. Oct 29, 2007 #4

    clem

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    Sorry, I should have had T=\sqrt{T^2+2MT}.
    I use T for KE, which is more common, and relativistic units with c absent.
    v_P=E/p, and v_g=dE/dp=p/E.
    You don't need gamma or v, but they are gamma=E/M and v=p/E.
     
  6. Oct 29, 2007 #5
    Buddy,what you are using seems not quite effective here...Remember we are to compare between group and phase velocities of an e and a p whose KE are the same...And you have not used the fact that their KE are the same...

    I am referring to another method...It is no approximation..stands on sheer logic...

    1 stands for e and 2 stands for p

    K=(γ₁-1)m₁c²=(γ₂-1)m₂c²
    Now,(γ₂-1)/(γ₁-1)=m₁/m₂

    =>(γ₂-γ₁)/(γ₁+γ₂-2)=(m₁-m₂)/(m₁+m₂)<0
    Also,(γ₁+γ₂)>2

    => γ₂<γ₁
    From which you can deduce the relation between group and phase velocity...
     
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